NOON State Generation with Phonons in Acoustic Wave Resonators

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The electronic ground state of the NV-center has a spin of S = 1 with an energy splitting between the states ms =0 | i and ms = 1 with frequency ω± 2π 2.88 GHz in a zero magnetic field. By applying an external magnetic field | ± i ≈ × B~ parallel to the axis between the nitrogen and the vacancy, the states ms = 1 can be split. For simplicity, the | ± i states of the NV-centers ms =0 , ms = 1 , and ms =1 are labelled as g , e , and u , respectively, with where | i | − i | i | i | i | i the energies are characterized by Eg < Ee and Eg < Eu. ωge (ωgu) is defined as the transition frequency between states g and e ( g and u ). Here, ωge can be tuned by B~ , and ωgu is kept unchanged as 2π 2.88 GHz. The | i | i | i | i × states g and e ( g and u ) of the NVE can be flipped by applying a microwave pulse with frequency ωe (ωu) and | i | i | i | i strength Ωe (Ωu) as shown in Fig.1(b). In addition, the frequencies of the AWRs and the transition g u of the NVE should be far detuned from each other largely. | i ↔ | i

There are two stages with (2N +2M + 3) steps to generate the NOON state. First, we focus on the ﬁrst stage which contains (2N + 1) steps. In this stage, the two transitions g e and g u of the NVE are set to be far | i ↔ | i | i ↔ | i detuned from AWR2 largely all the time. The initial state of the whole system should be prepared as

ψ = g 0 0 . (1) | iI | i | i1 | i2

Here, the subscript 1(2) represents the AWR1(2). The operations of the ﬁrst stage containing (2N + 1) steps are described as follows: Step 1: A microwave pulse with strength Ωe is applied to resonate with the transition g e of the NVE to form + + | i ↔ | i the Hamiltonian He = ~Ωe σ + H.c. , where σ represents the raising operator of the transition g e . Here, ge ge | i ↔ | i the two transitions of the NVE should be far detuned from AWR1 and AWR2 largely. Then, the state of the system will evolve from ψ to | iI 1 ψ = ( g i e ) 0 0 (2) | i1 √2 | i− | i | i1 | i2 after an operation time t = π/ (4Ωe). Step 2: A microwave pulse with strength Ωu is applied to resonate with the transition g u of the NVE when | i ↔ | i the two transitions of the NVE are detuned from AWR1 and AWR2. After an operation time of t = π/ (2Ωu), state g of the NVE is excited to u with a i phase shift, and the state of the whole system becomes | i | i − i ψ = − ( e + u ) 0 0 . (3) | i2 √2 | i | i | i1 | i2

Step 3: By letting the transition g e of the NVE resonate with AWR (ωge = ω ), state ψ will evolve to | i ↔ | i 1 1 | i2 i ψ = − ( i g 1 + u 0 ) 0 (4) | i3 √2 − | i | i1 | i | i1 | i2

ge ge after an operation time t = π/ (2g1 ). Here, ω1(2) is the frequency of the AWR1(2). g1 is the coupling strength between the AWR and the transition g e of the NVE. 1 | i ↔ | i 3

Step 4: A microwave pulse with strength Ωe is applied to excite the state g to e after an operation time of | i | i t = π/ (2Ωe). Then, the state of the system evolves from ψ to | i3 i ψ = − [ i ( i) e 1 + u 0 ] 0 . (5) | i4 √2 − − | i | i1 | i | i1 | i2

Step 5: As in the step 3, we let the transition g e of the NVE resonate with the AWR1. After an interaction time t = π/ (2gge), the state of the system evolves| i from ↔ | iψ to 1 | i4 i ψ = − [ i ( 1) g 2 + u 0 ] 0 . (6) | i5 √2 − − | i | i1 | i | i1 | i2 Step j (j = 6, 7, ..., 2N + 1): Repeating steps 4 and 5 successively with N + 1 times, the state of the system will evolve to

i N−1 ψ 2N+1 = − i ( 1) g N 1 + u 0 1 0 2 . (7) | i √2 h− − | i | i | i | i i | i

Then, by tuning the two transitions g e and g u of the NVE to be far detuned from AWR1 largely, we will give the operations of (2M + 2) steps| i ↔ of the| i second| i ↔stage | i to achieve the NOON state as follows: Step 2N + 2: A microwave pulse with strength Ωu is applied to ﬂip the states g and u of the NVE. After an | i | i operation time t = π/ (2Ωu), the state of the system becomes

′ 1 N−1 ψ 1 = − i ( 1) u N 1 0 2 + g 0 1 0 2 . (8) | i √2 h− − | i| i | i | i | i | i i ′ Step 2N + 3: A microwave pulse with strength Ωe is applied to excite the state g to e . State ψ will evolve to | i | i | i1

′ 1 N−1 ψ 2 = − i ( 1) u N 1 0 2 + ( i) e 0 1 0 2 . (9) | i √2 h− − | i | i | i − | i | i | i i

Step 2N + 4: Resonating the AWR2 with the transition g e of the NVE (ωge = ω2), the state of the system will evolve from ψ ′ to | i ↔ | i | i2

′ 1 N−1 ψ 3 = − i ( 1) u N 1 0 2 + ( 1) g 0 1 1 2 (10) | i √2 h− − | i | i | i − | i | i | i i ge ge after an operation time of t = π/ (2g2 ). Here, g2 is the coupling strength between the AWR2 and the transition g e of the NVE. | iThen, ↔ | i repeating the operations of steps 2N +3 and 2N + 4 with M 1 times, the state of the whole system evolves to − 1 ψ ′ = − i ( 1)N−1 u N 0 + ( 1)M−1 g 0 M 1 . (11) | i2M−1 √2 − − | i | i1 | i2 − | i | i1 | − i2

Step 2N +2M + 1: Applying a microwave pulse with strength Ωe to excite the state g to the state e , state ψ ′ will evolve to | i | i | i2M−1 1 ψ ′ = − i( 1)N−1 u N 0 + ( i) ( 1)M−1 e 0 M 1 . (12) | i2M √2 − − | i | i1 | i2 − − | i | i1 | − i2

Step 2N +2M +2: The same as the step 2N +2. The state of the system evolves from ψ ′ to | i2M 1 ψ ′ = − ( 1)N g N 0 + ( i) ( 1)M−1 e 0 M 1 . (13) | i2M+1 √2 − | i | i1 | i2 − − | i | i1 | − i2 Finally, resonating the transition g e of the NVE with the AWR , the state of the system becomes | i ↔ | i 2 1 ψ = − ( 1)N N 0 + ( 1)M 0 M g , (14) | if √2 − | i1 | i2 − | i1 | i2 | i which is just the NOON state of the AWRs. 4

III. NUMERICAL SIMULATION

The operations for generating the NOON state with the AWRs contain four kinds of resonant interactions: the microwave pulse with strength Ωe resonates with the transition g e of the NVE, the microwave pulse with | i ↔ | i strength Ωu resonates with the transition g u of the NVE, the transition g e of the NVE resonates with | i ↔ | i | i ↔ | i the AWR1, and the transition g e of the NVE resonates with the AWR2. Hamiltonians of the system for these interactions are given below: | i ↔ | i First, when the microwave pulse with strength Ωe resonates with the transition g e of the NVE, the Hamil- tonian of the system can be written as | i ↔ | i

u ge ge r ~ + ~ + i∆e t ~ + i∆1 t He = Ωe σge + H.c. + Ωe σgue + H.c. + g1 σgeb1e + H.c. gu ge gu ~ gu + i∆1 t ~ ge + i∆2 t ~ gu + i∆2 t + g1 σgu b1e + H.c. + g2 σgeb2e + H.c. + g2 σgub2e + H.c. . (15) + + Here, b1 (b2) is the annihilation operator of the AWR1 (AWR2), and σge (σgu) represents the raising operator of the gu gu transition g e ( g u ) of the NVE. g1 (g2 ) is the coupling strength between the AWR1 (AWR2) and the | i ↔ | i | ige ↔ | i ge gu gu u transition g u . ∆ = ωge ω , ∆ = ωge ω , ∆ = ωgu ω , ∆ = ωgu ω , and ∆ = ωgu ωe. ωe is | i ↔ | i 1 − 1 2 − 2 1 − 1 2 − 2 e − the frequency of the microwave pulse with strength Ωe. Second, the microwave pulse with strength Ωu is applied to resonate with the transition g u of the NVE. The Hamiltonian of the system can be expressed as | i ↔ | i

e ge ge r ~ + ~ + i∆ut ~ + i∆1 t Hu = Ωu σgu + H.c. + Ωu σgee + H.c. + g1 σgeb1e + H.c. gu ge gu ~ gu + i∆1 t ~ ge + i∆2 t ~ gu + i∆2 t + g1 σgub1e + H.c. + g2 σgeb2e + H.c. + g2 σgu b2e + H.c. , (16) e where ∆ = ωge ωu. u − Third, when the frequency of the transition g e of the NVE is tuned to resonate with AWR1, the Hamiltonian of the system becomes | i ↔ | i

gu ge gu r ~ ge + ~ gu + i∆1 t ~ ge + i∆2 t ~ gu + i∆2 t H1 = g1 σgeb1 +H.c. + g1 σgub1e +H.c. + g2 σgeb2e +H.c. + g2 σgub2e +H.c. . (17)

TABLE I: Parameters for generating a NOON state with N = M = 1.

step Ωe/2π(MHz) Ωu/2π(MHz) ωge/2π(GHz) (1) 1.4 0 2.7 (2) 0 1.3 2.7 (3) 0 0 0.1525 (4) 0 1.3 2.7 (5) 0.9 0 2.7 (6) 0 0.9 2.7 (7) 0 0 0.1848

TABLE II: Parameters for generating a NOON state with N = M = 2.

step Ωe/2π(MHz) Ωu/2π(MHz) ωge/2π(GHz) (1) 1.4 0 2.7 (2) 0 1.3 2.7 (3) 0 0 0.1525 (4) 0.9 0 2.7 (5) 0 0 0.1525 (6) 0 1.3 2.7 (7) 0.9 0 2.7 (8) 0 0 0.1848 (9) 0.9 0 2.7 (10) 0 0.9 2.7 (11) 0 0 0.1848 5

In the last case, resonating the frequency of the transition g e with the AWR2 and making the AWR2 far detuned from the transition g u of the NVE largely, the Hamiltonian| i ↔ | i can be expressed as | i ↔ | i gu ge gu r ~ ge + ~ gu + i∆2 t ~ ge + i∆1 t ~ gu + i∆1 t H2 = g2 σgeb2 +H.c. + g2 σgub2e +H.c. + g1 σgeb1e +H.c. + g1 σgub1e +H.c. . (18)

1 1 gge=ggu=0.32MHz 2 2 gge=ggu=0.28MHz 2 2 0.98 0.995 gge=ggu=0.35MHz 2 2 0.96 N=1 Fidelity Fidelity 0.99 N=2 0.94 N=3

0.92 0.985 0 200 400 600 1 2 3 φ φ (γ )−1=(γ )−1 (µs) Phonon number N e u (a) (b)

ge FIG. 2: (Color online) (a) The fidelities of the NOON states (N = M = 1, 2, and 3) with three different couplings: g2 = gu g2 = 2π × 0.32 MHz (blue square), 2π × 0.28 MHz (red circle), and 2π × 0.35 MHz (green triangle). (b) The fidelities of the NOON states vary with the dephasing rate of the NVE.

To show the feasibility of the scheme, we numerically simulate [95–97] the ﬁdelity of the NOON state through the r Hamiltonian Hj (j = e, u, 1, 2) by considering the relaxation rate and the dephasing rate of the NVE. The master equation governing the dynamics of the system is

dρ i r = H ,ρ + κ D [b ] ρ + κ D [b ] ρ + γgeD [σge] ρ + γguD [σgu] ρ dt −~ j 1 1 2 2 φ 1 1 φ 1 1 +γ σeeρσee σeeρ ρσee + γ σuuρσuu σuuρ ρσuu . (19) e − 2 − 2 u − 2 − 2

Here, κ1(2) is the decay rate of the AWR1(2). γge(γgu) is the energy relaxation rate of the transition g e φ φ | i ↔ | i ( g u ) of the NVE, and γe (γu ) is the dephasing rate of the state e ( u ). σee = e e and σuu = u u . | i ↔ | i † † † | i | i | i h | | i h | D [O] ρ = (2OρO O Oρ ρO O)/2 with O = b , b , σge, and σgu. The ﬁdelity of the NOON state is deﬁned as − − 1 2

F =f ψ ρ(t) ψ . (20) h | | if r Here, ρ(t) is the realistic density operator after the operations on the initial state ψ I with the Hamiltonian Hj and the decoherence of the system. ψ is the ﬁnal state after the ideal operations on| thei initial state ψ . | if | iI

TABLE III: Parameters for generating a NOON state with N = M = 3.

step Ωe/2π(MHz) Ωu/2π(MHz) ωge/2π(GHz) (1) 1.4 0 2.7 (2) 0 1.3 2.7 (3) 0 0 0.1525 (4) 0.9 0 2.7 (5) 0 0 0.1525 (6) 0.9 0 2.7 (7) 0 0 0.1525 (8) 0 1.3 2.7 (9) 0.9 0 2.7 (10) 0 0 0.1848 (11) 0.9 0 2.7 (12) 0 0 0.1848 (13) 0.9 0 2.7 (14) 0 0.9 2.7 (15) 0 0 0.1848 6

Here, the parameters of the system are taken as: ω1 =2π 152.5 MHz, ω2 =2π 184.8 MHz [64], ωe =2π 2.7 ×ge gu ge gu × × GHz, and ωu = 2π 2.88 GHz. The couplings are taken as g1 = g1 = g2 = g2 = 2π 0.32 MHz, which means × × 5 the coupling strength between a single NV and the AWR is gs/2π 1 kHz [98] when there are 10 NV-centers in ∼ −1 −1 the ensemble. κ−1 = κ−1 = 9.83 102 s [64], γ−1 = γ−1 = 6 ms, γφ = γφ = 600 µs [99]. The remaining 1 2 × ge gu e u parameters in each step for generating the NOON states with N = M = 1, 2, and 3 are shown in Table I, Table II, and Table III, respectively. The fidelities of our NOON states plotted in Fig.2(a) indicate the fidelities of the NOON states with N = M = 1, 2, and 3 reach F1 = 99.54%, F2 = 99.18%, F3 = 98.80%, respectively. To discuss the influences of the imperfect relationship among parameters, for simplicity, we consider two conditions: (1) coupling strengths between each transitions of the NVE and the two AWRs are not equal to each other; (2) the ge gu ge gu influences of different dephasing rates of the NVE. To consider the first condition, we take g1 = g1 >g2 = g2 and ge gu ge gu ge gu ge gu g1 = g1

IV. SUMMARY

We propose a scheme to generate a NOON state using AWRs in a system consisting of two AWRs coupled to an NVE. With the resonant interactions between the AWR (microwave pulse) and the NVE, the numerical simulation shows that the ﬁdelities of our NOON states can reach 99.54% for N = 1, 99.18% for N = 2, and 98.80% for N =3 by considering decoherence of the system.

ACKNOWLEDGEMENTS

M. Hua was supported by the National Natural Science Foundation of China under Grants No. 11704281 and No. 11647042.

[1] A. K. Ekert, Phys. Rev. Lett. 1991, 67, 661. [2] M. Hillery, V. Buˇzek, A. Berthiaume, Phys. Rev. A 1999, 59, 1829. [3] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett. 1993, 70, 1895. [4] G. L. Long, X. S. Liu, Phys. Rev. A 2002, 65, 032302. [5] C. H. Bennett, S. J. Wiesner, Phys. Rev. Lett. 1992, 69, 2881. [6] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK 2000. [7] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 2009, 81, 865. [8] G. Castagnoli, Found. Phys. 2016, 46, 360. [9] F. G. Deng, B. C. Ren, X. H. Li, Sci. Bull. 2017, 62, 46. [10] Y. B. Sheng, L. Zhou, Sci. Bull. 2017, 62, 1025. [11] H. K. Lo, M. Curty, B. Qi, Phys. Rev. Lett. 2012, 108, 130503. [12] P. H. Niu, Z. R. Zhou, Z. S. Lin, Y. B. Sheng, L. G. Yin, G. L. Long, Sci. Bull. 2018, 63, 1345. [13] T. Li, Z. Q. Yin, Sci. Bull. 2016, 61, 163. [14] Y. B. Sheng, J. Pan, R. Guo, L. Zhou, L. Wang, Sci. China: Phys., Mech. Astron. 2015, 58, 060301. [15] I. Buluta, S. Ashhab, F. Nori, Rep. Prog. Phys. 2011, 74, 104401. [16] Z. L. Xiang, S. Ashhab, J. Q. You, F. Nori, Rev. Mod. Phys. 2013, 85, 623. [17] Z. Y. Xue, Z. D. Wang, Phys. Rev. A 2007, 75, 064303. [18] D. Ballester, Phys. Rev. A 2009, 79, 062317. [19] C. P. Yang, Phys. Rev. A 2011, 83, 062302. [20] P. B. Li, F. L. Li, Opt. Express 2011, 19, 1207. [21] Z. Jin, S. L. Su, A. D. Zhu, H. F. Wang, S. Zhang, Opt. Express 2017, 25, 88. [22] X. Q. Shao, J. H. Wu, X. X. Yi, G. L. Long, Phys. Rev. A 2017, 96, 062315. [23] W. Qin, A. Miranowicz, P. B. Li, X. Y. L¨u, J. Q. You, F. Nori, Phys. Rev. Lett. 2018, 120, 093601. [24] J. Q. You, F. Nori, Phys. Today 2005, 58, 42. [25] J. Q. You, F. Nori, Nature 2011, 474, 589. [26] X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu, F. Nori, Phys. Rep. 2017, 718 -719, 1-102. 7

[27] M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’ Connell, D. Sank, H. Wang, M. Weides, J.Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, J. M. Martinis, Nature (London) 2010, 467, 570. [28] L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, R. J. Schoelkopf, Nature (London) 2010, 467, 574. [29] Z. Liu, L. Kuang, K. Hu, L. Xu, S. Wei, L. Guo, X. Q. Li, Phys. Rev. A 2010, 82, 032335. [30] Y. Hu, L. Tian, Phys. Rev. Lett. 2011, 106, 257002. [31] P. B. Li, S. Y. Gao, F. L. Li, Phys. Rev. A 2013, 88, 043802. [32] S. L. Ma, Z. Li, A. P. Fang, P. B. Li, S. Y. Gao, F. L. Li, Phys. Rev. A 2014, 90, 062342. [33] J. R. Johansson, N. Lambert, I. Mahboob, H. Yamaguchi, F. Nori, Phys. Rev. B 2014, 90, 174307. [34] C. P. Yang, Q. P. Su, S. B. Zheng, F. Nori, S. Han, Phys. Rev. A 2017, 95, 052341. [35] Z. P. Yang, Z. Li, S. L Ma, F. L Li, Phys. Rev. A 2017, 96, 012327. [36] L. M. Duan, C. Monroe, Rev. Mod. Phys. 2010, 82, 1209. [37] T. Dutta, M. Mukherjee, K. Sengupta, Phys. Rev. Lett. 2013, 111, 170406. [38] S. Zippilli, M. Johanning, S. M. Giampaolo, Ch. Wunderlich, F. Illuminati, Phys. Rev. A 2014, 89, 042308. [39] J. S. Pedernales, R. Di Candia, P. Schindler, T. Monz, M. Hennrich, J. Casanova, E. Solano, Phys. Rev. A 2014, 90, 012327. [40] T. J. Wang, Y. Lu, G. L. Long, Phys. Rev. A 2012, 86, 042337. [41] J. R. Schaibley, A. P. Burgers, G. A. McCracken, L. M. Duan, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, L. J. Sham, Phys. Rev. Lett. 2013, 110, 167401. [42] J. Hakami, M. Suhail Zubairy, Phys. Rev. A 2016, 93, 022320. [43] R. Bonazzola, J. A. Andrade, J. I. Facio, D. J. Garc´ıa, P. S. Cornaglia, Phys. Rev. B 2017, 96, 075157. [44] Y. L. Delley, M. Kroner, S. Faelt, W. Wegscheider, A. Imamo˘glu,˙ Phys. Rev. B 2017, 96, 241410. [45] M. Müller, H. Vural, C. Schneider, A. Rastelli, O. G. Schmidt, S. Höfling, P. Michler, Phys. Rev. Lett. 2017, 118, 257402. [46] X. D. Tian, Y. M. Liu, C. L. Cui, J. H. Wu, Phys. Rev. A 2015, 92, 063411. [47] J. C. Lee, K. K. Park, T. M. Zhao, Y. H. Kim, Phys. Rev. Lett. 2016, 117, 250501. [48] D. D. Bhaktavatsala Rao, S. Yang, J. Wrachtrup, Phys. Rev. B 2015, 92, 081301. [49] E. K. Levi, P. G. Kirton, B. W. Lovett, Phys. Rev. A 2016, 94, 032302. [50] F. Dolde, I. Jakobi, B. Naydenov, N. Zhao, S. Pezzagna, C. Trautmann, J. Meijer, P. Neumann, F. Jelezko, J. Wrachtrup, Nat. Phys. 2013, 9, 139. [51] X. Han, Q. Guo, A. D. Zhu, S. Zhang, H. F. Wang, Opt. Express 2017, 25, 17701. [52] D. D. Bhaktavatsala Rao, S. Yang, Jörg Wrachtrup, Phys. Rev. A 2017, 95, 022310. [53] J. A. Jones, M. Mosca, R. H. Hansen, Nature 1998, 393, 344. [54] G. L. Long, L. Xiao, J. Chem. Phys. 2003, 119, 8473. [55] G. R. Feng, G. F. Xu, G. L. Long, Phys. Rev. Lett. 2013, 110, 190501. [56] T. Xin, J. S. Pedernales, E. Solano, G. L. Long, Phys. Rev. A 2018, 97, 022322. [57] Q. Yu, Y. B. Zhang, J. Li, H. Y. Wang, X. H. Peng, J. F. Du, Sci. China: Phys., Mech. Astron. 2017, 60, 070313. [58] G. Giavaras, J. H. Jefferson, A. Ramˇsak, T. P. Spiller, C. J. Lambert, Phys. Rev. B 2006, 74, 195341. [59] L. Tian, Phys. Rev. Lett. 2013, 110, 233602. [60] Y. D. Wang, A. A. Clerk, Phys. Rev. Lett. 2013, 110, 253601. [61] J. Q. Liao , Q. Q. Wu, F. Nori, Phys. Rev. A 2014, 89, 014302. [62] M. Gao, F. C. Lei, C. G. Du, G. L. Long, Sci. China: Phys., Mech. Astron. 2016, 59, 610301. [63] K. Hammerer, A. S. Sørensen, E. S. Polzik, Rev. Mod. Phys. 2010, 82, 1041. [64] N. Wang, J. M. Tsai, F. L. Hsiao, B. W. Soon, D. L. Kwong, M. Palaniapan, C. Lee, IEEE Electron Device Lett. 2011, 32, 821. [65] M. Goryachev, D. L. Creedon, E. N. Ivanov, S. Galliou, R. Bourquin, M. E. Tobar, Appl. Phys. Lett. 2012, 100, 243504. [66] J. R. Gell, M. B. Ward, R. J. Young, R. M. Stevenson, P. Atkinson, D. Anderson, G. A. C. Jones, D. A. Ritchie, A. J. Shields, Appl. Phys. Lett. 2008, 93, 081115. [67] M. J. A. Schuetz, E. M. Kessler, G. Giedke, L. M. K. Vandersypen, M. D. Lukin, J. I. Cirac, Phys. Rev. X 2015, 5, 031031. [68] D. Andrew Golter, Thein Oo, M. Amezcua, I. Lekavicius, K. A. Stewart, H. Wang, Phys. Rev. X 2016, 6, 041060. [69] D. Andrew Golter, Thein Oo, M. Amezcua, K. A. Stewart, H. Wang, Phys. Rev. Lett. 2016, 116, 143602. [70] K. Cai, Z. W. Pan, R. X. Wang, D. Ruan, Z. Q. Yin, G. L. Long, Opt. Lett. 2018, 43, 1163. [71] R. Manenti, M. J. Peterer, A. Nersisyan, E. B. Magnusson, A. Patterson, P. J. Leek, Phys. Rev. B 2016, 93, 041411. [72] A. Noguchi, R. Yamazaki, Y. Tabuchi, Y. Nakamura, Phys. Rev. Lett. 2017, 119, 180505. [73] R. J. Jiménez Riobóo, A. Sánchez-Sánchez, C. Prieto, Phys. Rev. B 2016, 94, 014313. [74] F. Iikawa, A. Hernández-M´ınguez, M. Ramsteiner, P. V. Santos, Phys. Rev. B 2016, 93, 195212. [75] G. Michel, F. Pétrélis, S. Fauve, Phys. Rev. Lett. 2016, 116, 174301. [76] M. Kervinen, I. Rissanen, M. Sillanpää, Phys. Rev. B 2018 97, 205443. [77] A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J.M. Martinis, A. N. Cleland, Nature (London) 2010, 464, 697. [78] J. M. Pirkkalainen, S. U. Cho, J. Li, G. S. Paraoanu, P. J. Hakonen, M. A. Sillanpää, Nature (London) 2013, 494, 211. [79] M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekstrom, G. Johansson, P. Delsing, Science 2014, 346, 207. [80] Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, R. J. Schoelkopf, Science 2017, 358, 199. [81] R. Manenti, A. F. Kockum, A. Patterson, T. Behrle, J. Rahamim, G. Tancredi, F. Nori, P. J. Leek, Nat. Commun. 2017, 8

8, 975. [82] A. F. Kockum, G. Johansson, F. Nori, Phys. Rev. Lett. 2018, 120, 140404. [83] P. B. Li, Z. L. Xiang, P. Rabl, F. Nori, Phys. Rev. Lett. 2016, 117, 015502. [84] C. S. Mu˜noz, A. Lara, J. Puebla, F. Nori, Phys. Rev. Lett. 2018, 121, 123604. [85] O. O. Soykal, R. Ruskov, C. Tahan, Phys. Rev. Lett. 2011, 107, 235502. [86] J. R. Gell, M. B. Ward, R. J. Young, R. M. Stevenson, P. Atkinson, D. Anderson, G. A. C. Jones, D. A. Ritchie, A. J. Shields, Appl. Phys. Lett. 2008, 93, 081115. [87] O. D. D. Couto, S. Lazic, F. Iikawa, J. A. H. Stotz, U. Jahn, R. Hey, P. V. Santos, Nat. Photonics 2009, 3, 645. [88] M. Metcalfe, S. M. Carr, A. Muller, G. S. Solomon, J. Lawall, Phys. Rev. Lett. 2010, 105, 037401. [89] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, R. L. Walsworth, Nat. Commun. 2013, 4, 1743. [90] X. Y. L¨u, Z. L. Xiang, W. Cui, J. Q. You, F. Nori, Phys. Rev. A 2013, 88, 012329. [91] N. Zhao, Z. Q. Yin, Phys. Rev. A 2014, 90, 042118. [92] H. R. Wei, G. L. Long, Phys. Rev. A 2015, 91, 032324. [93] Y. Ma, T. M. Hoang, M. Gong, T. Li, Z. Q. Yin, Phys. Rev. A 2017, 96, 023827. [94] P. B. Li, F. Nori, Phys. Rev. Appl. 2018, 10, 024011. [95] I. Buluta, F. Nori, Science 2009, 326, 108. [96] I. M. Georgescu, S. Ashhab, F. Nori, Rev. Mod. Phys. 2014, 86, 153. [97] N. Shammah, S. Ahmed, N. Lambert, S. D. Liberato, F. Nori, arXiv: 1805.05129, 2018. [98] S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, M. D. Lukin, Phys. Rev. Lett. 2013, 110, 156402. [99] M. J. Tao, M. Hua, Q. Ai, F. G. Deng, Phys. Rev. A 2015, 91, 062325.